General sum-connectivity index of trees with given number of branching vertices

Document Type : Research Paper

Author

Department of Mathematics and Applied Mathematics, University of the Free State, Bloemfontein, South Africa

Abstract

In 2015, Borovi'{c}anin presented trees with the smallest first Zagreb index among trees with given number of vertices and number of branching vertices. The first Zagreb index is obtained from the general sum-connectivity index if $a = 1$. For $a \in \mathbb{R}$, the general sum-connectivity index of a graph $G$ is defined as $\chi_{a} (G) = \sum_{uv\in E(G)} [d_G (u) + d_G (v)]^{a}$, where $E(G)$ is the edge set of $G$ and $d_G (v)$ is the degree of a vertex $v$ in $G$. We show that the result of Borovi'{c}anin cannot be generalized for the general sum-connectivity index ($\chi_{a}$ index) if $0 < a < 1$ or $a > 1$. Moreover, the sets of trees having the smallest $\chi_a$ index are not the same for $0 < a < 1$ and $a > 1$. Among trees with given number of vertices and number of branching vertices, we present all the trees with the smallest $\chi_a$ index for $0 < a < 1$ and $a > 1$. Since the hyper-Zagreb index is obtained from the $\chi_a$ index if $a = 2$, results on the hyper-Zagreb index are corollaries of our results on the $\chi_a$ index for $a > 1$.

Keywords

Main Subjects

References

[1] M. Aghel, A. Erfanian and A.R. Ashrafi, On the first and second Zagreb indices of quasi unicyclic graphs, Trans.
Comb. 8 no. 3 (2019) 29–39.
[2] S. Ahmed, On the maximum general sum-connectivity index of trees with a fixed order and maximum degree, Discrete Math. Algorithms Appl. 13 no. 4 (2021) 13 p.
[3] A. Ali, L. Zhong and I. Gutman, Harmonic index and its generalizations: Extremal results and bounds, MATCH
Commun. Math. Comput. Chem. 81 no. 2 (2019) 249–311.
[4] B. Borovićanin, On the extremal Zagreb indices of trees with given number of segments or given number of branching vertices, MATCH Commun. Math. Comput. Chem. 74 no. 1 (2015) 57–79.
[5] Q. Cui and L. Zhong, On the general sum-connectivity index of trees with given number of pendent vertices, Discrete
Appl. Math. 222 (2017) 213–221.
[6] Z. Du, B. Zhou and N. Trinajstić, On the general sum-connectivity index of trees, Appl. Math. Lett. 24 no. 3 (2011)
402–405.
[7] I. Gutman, D. Hanyuan, S. Balachandran, S.K. Ayyaswamy and Y.B. Venkatakrishnan, On the average eccentricity,
the harmonic index and the largest signless Laplacian eigenvalue of a graph, Trans. Comb. 6 no. 4 (2017) 43–50.
[8] M.K. Jamil and I. Tomescu, General sum-connectivity index of trees and unicyclic graphs with fixed maximum degree,
Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci. 20 no. 1 (2019) 11–17.
[9] M.K. Jamil and I. Tomescu, Minimum general sum-connectivity index of trees and unicyclic graphs having a given
matching number, Discrete Appl. Math. 222 (2017) 143–150.
[10] M. Liang, B. Cheng and J. Liu, Solution to the minimum harmonic index of graphs with given minimum degree,
Trans. Comb. 7 no. 2 (2018) 25–33.
[11] H. Liu and Z. Tang, Maximal hyper-Zagreb index of trees, unicyclic and bicyclic graphs with a given order and
matching number, Discrete Math. Lett. 4 (2020) 11–18.
[12] I. Tomescu and M.K. Jamil, Maximum general sum-connectivity index for trees with given independence number,
MATCH Commun. Math. Comput. Chem. 72 no. 3 (2014) 715–722.
[13] I. Tomescu and S. Kanwal, Ordering trees having small general sum-connectivity index, MATCH Commun. Math.
Comput. Chem. 69 no. 3 (2013) 535–548.
[14] T. Vetrı́k, General Randić index of trees with given number of branching vertices, Discrete Math. Algorithms Appl.,
in press.
[15] T. Vetrı́k and M. Masre, Generalized eccentric connectivity index of trees and unicyclic graphs, Discrete Appl. Math.
284 (2020) 301–315.
[16] B. Zhou and N. Trinajstić, On general sum-connectivity index, J. Math. Chem. 47 (2010) 210–218.
[17] L. Zhong and Q. Qian, The minimum general sum-connectivity index of trees with given matching number, Bull.
Malays. Math. Sci. Soc., 43 no. 2 (2020) 1527–1544.
[18] Z. Zhu and W. Zhang, Trees with a given order and matching number that have maximum general sum-connectivity
index, Ars Combin., 128 (2016) 439–446.
Transactions on Combinatorics
Volume 12, Issue 4 - Serial Number 4
December 2023
Pages 227-238

Files

Cited by

History

  • Receive Date: 07 May 2022
  • Revise Date: 11 October 2022
  • Accept Date: 21 October 2022
  • Published Online: 01 December 2023

Share

How to cite

Statistics